Explicit exponential decay $|J_n|\le 10e^{-7n}$ of the even-index Zeilberger–Zudilin integrals

For every $n\ge 0$,

where $J_n=i\int_{-1-2i}^{-1+2i}R_n(t)\,dt$ and $R_n(t)=5t^{4n}(t^4+6t^2+25)^{4n}(25-t^2)^{-6n-1}$, as in the definition file PiIrrationality_ZZEvenForms.

Zeilberger and Zudilin show $\limsup|I_m|^{1/m}=|N_1|=0.02945849\ldots$ for their integrals $I_m$. At even index $m=2n$ the true rate is $|N_1|^2=0.000867803\ldots<e^{-7}=0.000911882\ldots$. The rate is attained along a contour inside the disc $|t|<5$ that passes through the saddle points of $|t|^4|t^4+6t^2+25|^4/|25-t^2|^6$. The statement is an explicit, slightly weaker form with a uniform constant.